#452329

order	512
nilpotency class	3
isoclinism	isoclinic class of #417903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

791 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,4)(3,7,8,6)(9,10,14,17,15,20,13,19)(11,21,24,16,25,31,23,33)(12,26,28,34,30,18,29,36)(22,37,27,32,39,35,38,40);(1,3)(2,6)(4,7)(5,8)(9,11,15,25)(10,16,20,33)(12,27,30,38)(13,24,14,23)(17,21,19,31)(18,35,26,37)(22,29,39,28)(32,34,40,36);(1,4,5,2)(3,7,8,6)(10,17,20,19)(16,31,33,21)(18,34,26,36)(32,37,40,35);(3,8)(6,7)(9,12,14,29,15,30,13,28)(10,18,19,36,20,26,17,34)(11,22,24,38,25,39,23,27)(16,32,21,35,33,40,31,37)

Polycyclic presentation:

2^9|a^2=egh,b^2=i,c^2=ef,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359525);